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Concepts Of Physics MCQ Edition [Volume 2]
Exam
Concepts Of Physics MCQ Edition [Volume 2]
Subject
Physics
Chapter
Semiconductors and Semiconductor Devices

Concepts Of Physics MCQ Edition [Volume 2] · Physics · Semiconductors and Semiconductor Devices

The temperature dependence of the conductivity of an intrinsic...

The temperature dependence of the conductivity of an intrinsic semiconductor is given by σ=σ0e−ΔE/2kT\sigma = \sigma_0 e^{-\Delta E/2kT}, where σ0\sigma_0 is a constant. Determine the temperature at which the conductivity of an intrinsic germanium semiconductor becomes twice its value at T=300 KT = 300 \text{ K}. Assume the energy gap for germanium is 0.650 eV0.650 \text{ eV} and it stays constant as temperature increases.

Options

  1. A336 K336 \text{ K}
  2. B308 K308 \text{ K}
  3. C318 K318 \text{ K}
  4. D284 K284 \text{ K}

Correct answer

C. 318 K318 \text{ K}

Step-by-step solution

The conductivity at temperature TT is given by σ=σ0e−ΔE/2kT\sigma = \sigma_0 e^{-\Delta E/2kT}.

Let the initial temperature be T1=300 KT_1 = 300 \text{ K} and the final temperature be T2T_2.

Given that σ2=2σ1\sigma_2 = 2\sigma_1, we have:

σ0e−ΔE/2kT2=2σ0e−ΔE/2kT1\sigma_0 e^{-\Delta E/2kT_2} = 2\sigma_0 e^{-\Delta E/2kT_1}

Taking the natural logarithm on both sides:

−ΔE2kT2=ln⁡2−ΔE2kT1-\dfrac{\Delta E}{2kT_2} = \ln 2 - \dfrac{\Delta E}{2kT_1}

1T2=1T1−2kln⁡2ΔE\dfrac{1}{T_2} = \dfrac{1}{T_1} - \dfrac{2k \ln 2}{\Delta E}

Substituting the given values T1=300 KT_1 = 300 \text{ K}, k=8.625×10−5 eV/Kk = 8.625 \times 10^{-5} \text{ eV/K}, ΔE=0.650 eV\Delta E = 0.650 \text{ eV}, and ln⁡2=0.693\ln 2 = 0.693:

1T2=1300−2×8.625×10−5×0.6930.650\dfrac{1}{T_2} = \dfrac{1}{300} - \dfrac{2 \times 8.625 \times 10^{-5} \times 0.693}{0.650}

1T2=3.333×10−3−1.839×10−4\dfrac{1}{T_2} = 3.333 \times 10^{-3} - 1.839 \times 10^{-4}

1T2=3.149×10−3 K−1\dfrac{1}{T_2} = 3.149 \times 10^{-3} \text{ K}^{-1}

T2=13.149×10−3≈317.5 KT_2 = \dfrac{1}{3.149 \times 10^{-3}} \approx 317.5 \text{ K}

Rounding to the nearest integer, we get T2≈318 KT_2 \approx 318 \text{ K}.
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